gen.tex

\documentclass[border=3pt,tikz]{standalone}
\usepackage{tikz}

\makeatletter

\def\venn@strip#1#2\venn@STOP{%
  \def\venn@next{#1}%
  \gdef\venn@rest{#2}%
}

\newcommand{\venn}[1]{%
\begin{tikzpicture}
\coordinate (A) at (0,0);
\coordinate (B) at (2,0);
\coordinate (C) at (1,{sqrt(3)});
\coordinate (S-SE) at (5,-3);
\coordinate (S-NW) at (-3,{sqrt(3)+3});
  \edef\venn@rest{#100000000}%
  \foreach \i in {0,...,7} {
  \begin{scope}[even odd rule]
    \expandafter\venn@strip\venn@rest\venn@STOP
    \ifnum\venn@next=1\relax
    \pgfmathparse{Mod(\i,2) == 1 ? "(S-SE) rectangle (S-NW)" : ""}
    \path[clip] \pgfmathresult (A) circle[radius=2];
    \pgfmathparse{Mod(floor(\i/2),2) == 1 ? "(S-SE) rectangle (S-NW)" : ""}
    \path[clip] \pgfmathresult (B) circle[radius=2];
    \pgfmathparse{Mod(floor(\i/4),2) == 1 ? "(S-SE) rectangle (S-NW)" : ""}
    \path[clip] \pgfmathresult (C) circle[radius=2];
    \fill[rounded corners,gray] (S-SE) rectangle (S-NW);
    \fi
  \end{scope}
  }
    \draw[ultra thick] (A) circle[radius=2];
    \draw[ultra thick] (B) circle[radius=2];
    \draw[ultra thick] (C) circle[radius=2];
    \node at (1,3) {$A$};
    \node at (-1,-1) {$B$};
    \node at (3,-1) {$C$};
    \draw[ultra thick,rounded corners] (S-SE) rectangle (S-NW);
\end{tikzpicture}
\newpage
}

\makeatother

\newcommand{\allvendiagrams}{
% To generate the lot:
\foreach \j in {0,...,255} {
  \def\venncode{}
  \foreach \k in {0,...,7} {
    \pgfmathparse{Mod(floor(\j/2^\k),2) == 1 ? "\venncode1" : "\venncode0"}
    \global\let\venncode=\pgfmathresult
  }
  \venn{\venncode}

}
}

\begin{document}
\allvendiagrams
\end{document}